Relational encryption for password verification

ABSTRACT

A method of equality verification using relational encryption including receiving a relational key that includes a first relational key component and a registration ciphertext that includes an encryption of a first plaintext data set. The method includes storing the registration ciphertext without decrypting the registration ciphertext. After the storing of the registration ciphertext, the method includes receiving an authentication request and communicating a safeguard data set that includes a random challenge in response to the authentication request. The method includes receiving an encrypted response that is generated based on the safeguard data set and a second plaintext data set. The method includes verifying a relationship between the encrypted response and the registration ciphertext using the relational key without decrypting the encrypted response and without decrypting the registration ciphertext. The relationship indicates that equality exists between the first and the second plaintext data sets.

FIELD

The embodiments discussed herein are related to equality verification using relational encryption.

BACKGROUND

User verification often includes equality verification. Generally, equality verification includes a system receiving a first data set. Later the system receives a second data set. The system performs a comparison to determine whether or not the first data set and the second data set are equal.

The subject matter claimed herein is not limited to embodiments that solve any disadvantages or that operate only in environments such as those described above. Rather, this background is only provided to illustrate one example technology area where some embodiments described herein may be practiced.

SUMMARY

According to an aspect of an embodiment, a method of equality verification using relational encryption includes receiving, from a trusted entity server, a relational key that may include a first relational key component. The method may include receiving, from a user computing system, a registration ciphertext that may include an encryption of a first plaintext data set. The method may include storing, in a non-transitory computer-readable storage media, the registration ciphertext without decrypting the registration ciphertext. After the storing of the registration ciphertext, the method may include receiving, from the user computing system, an authentication request. In response to the authentication request, the method may include communicating a safeguard data set that includes a random challenge to the user computing system. The method may include receiving, from the user computing system, an encrypted response that is generated based at least partially on the safeguard data set and a second plaintext data set. The method may include verifying, by the one or more processors, a relationship between the encrypted response and the registration ciphertext using the relational key. The verifying occurs without decrypting the encrypted response and without decrypting the registration ciphertext, the relationship may indicate that equality exists between the first plaintext data set and the second plaintext data set.

The object and advantages of the embodiments will be realized and achieved at least by the elements, features, and combinations particularly pointed out in the claims.

It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory and are not restrictive of the invention, as claimed.

BRIEF DESCRIPTION OF THE DRAWINGS

Example embodiments will be described and explained with additional specificity and detail through the use of the accompanying drawings in which:

FIG. 1 is a block diagram of an example operating environment;

FIG. 2 illustrates an example equality verification process in the operating environment of FIG. 1;

FIG. 3 illustrates an example computing system configured for equality verification using relational encryption;

FIG. 4 is a flow diagram of a method of equality verification using relational encryption; and

FIG. 5 is a flow diagram of another method of equality verification using relational encryption,

all arranged in accordance with at least one embodiment described herein.

DESCRIPTION OF EMBODIMENTS

User verification often includes equality verification. Generally, equality verification includes a system receiving a first data set. Later the system receives a second data set. The system performs a comparison to determine whether or not the first data set and the second data set are equal. In some systems, the first data set and the second data set may be encrypted when communicated to the system. To perform the comparison, the system decrypts the first data set and the second data set. The decryption often involves a private key stored on the system or a device controlled by the system. Moreover, the decryption involves storage, at least temporarily, of plaintext versions of the first data set and/or the second data set.

Accordingly, systems that perform equality verification in this or a similar fashion expose the system and users of the system to multiple vulnerabilities. For instance, an adversary may hack the system and access decrypted versions of the first data set and/or the second data set. Additionally or alternatively, an adversary may access the private key and intercept communications of the first data set and/or the second data set. The adversary may accordingly have access to the plaintext versions of the first data set and/or the second data set. Furthermore, if the system becomes an adversary, the system has access to the plaintext versions of the first data set and/or the second data set.

Some example embodiments described in this disclosure relate to systems and methods of equality verification using relational encryption. In relational encryption, instead of a comparison of decrypted data sets, equality is based on a relationship between encrypted data sets. Thus, a system implementing equality verification using relational encryption may not access or ever “know” a plaintext version of an encrypted data set. Furthermore, in relational encryption, the relationships between the encrypted data sets are based on a relational key. The relational key does not allow a plaintext data set to be derived from a ciphertext version of plaintext data set. Instead, the relational key may only allow a determination of whether the relationship exists. The relationship between the encrypted data sets is indicative of equality of the plaintext versions of the encrypted data sets.

Some additional details of these and other embodiments are discussed with respect to the appended figures in which commonly labeled items indicate similar structures unless described otherwise. The drawings are diagrammatic and schematic representations of some embodiments, and are not meant to be limiting, nor are they necessarily drawn to scale. Throughout the drawings, like numbers generally reference like structures unless described otherwise.

FIG. 1 illustrates a block diagram of an example operating environment 100, arranged in accordance with at least one embodiment described herein. In the operating environment 100, authentication of a user 106 may be performed by an authentication server 140. The authentication may include an equality verification between two plaintext data sets based on two corresponding ciphertext data sets that are communicated to the authentication server 140 at different times. The authentication may be performed using relational encryption.

Generally, relational encryption enables the authentication server 140 to ascertain relationships between ciphertext data sets that are indicative of equality between the plaintext data sets, without decrypting the ciphertext data sets. Moreover, relational encryption enables the authentication without storage of plaintext versions of the ciphertext data sets, without granting the authentication server 140 access to the plaintext versions, and without providing to the authentication server 140 one or more keys used to encrypt the plaintext data sets.

For example, the user 106 may communicate a registration ciphertext to the authentication server 140. The registration ciphertext may be representative of a first plaintext data set. An example of the first plaintext data set may include a password or another user-identifying data set. The registration ciphertext may be stored in a database 116 without decrypting or otherwise ascertaining the first plaintext data set represented by the registration ciphertext.

After the registration ciphertext is stored in the database 116, the user 106 may communicate an encrypted response to the authentication server 140. The encrypted response may be representative of a second plaintext data set. Using a relational encryption protocol, the authentication server 140 may determine whether one or more relationships exist between the encrypted response and the registration ciphertext. The relationship may be indicative of equality between the first plaintext data set and the second plaintext data set. The determination may be performed without decryption of the encrypted response or the registration ciphertext. Moreover, the relationship may be determined using a relational key that does not enable decryption of the encrypted response and/or the registration ciphertext. Some additional details of the relational encryption protocol are provided elsewhere in this disclosure.

The operating environment 100 includes a user computing system 104 that may be associated with the user 106, the database 116, a trusted entity server 108, the authentication server 140, and a network 107. The user computing system 104, the database 116, the trusted entity server 108, the authentication server 140 (collectively, “environment components”) may be communicatively coupled via the network 107. The environment components may communicate data and information used to authenticate the user 106 or one or more data sets via the network 107. Each of the environment components are briefly described in the following paragraphs.

The network 107 may include a wired network, a wireless network, or any combination thereof. The network 107 may include any suitable configuration or configurations including a star configuration, token ring configuration, or other configurations. The network 107 may include a local area network (LAN), a wide area network (WAN) (e.g., the Internet), and/or other interconnected data paths across which multiple devices may communicate. In some embodiments, the network 107 may include a peer-to-peer network. The network 107 may also be coupled to or include portions of a telecommunications network that may enable communication of data in a variety of different communication protocols. In some embodiments, the network 107 includes BLUETOOTH® communication networks and/or cellular communication networks for sending and receiving data including via short messaging service (SMS), multimedia messaging service (MMS), hypertext transfer protocol (HTTP), direct data connection, wireless application protocol (WAP), e-mail, or the like.

The trusted entity server 108 may include a processor-based computing system. For example, the trusted entity server 108 may include a hardware server or another processor-based computing system configured to function as a server. The trusted entity server 108 may include memory and network communication capabilities. In the operating environment 100, the trusted entity server 108 may be configured to communicate with the user computing system 104, the authentication server 140, and the database 116 via the network 107.

The trusted entity server 108 may be associated with a trusted entity. For example, the trusted entity may include a non-interested third party such as a certification authority. The user 106 and an entity associated with the authentication server 140 may trust, select, and agree upon the trusted entity.

The trusted entity server 108 may include a key generation module 118. The key generation module 118 may be configured to generate keys used in a relational encryption protocol. In some embodiments, the keys may include a public key set and a relational key. The keys generated by the key generation module 118 may be communicated to the user computing system 104 and the authentication server 140 or made available via the network 107.

The key generation module 118 may be implemented using hardware including a processor, a microprocessor (e.g., to perform or control performance of one or more operations), a field-programmable gate array (FPGA), or an application-specific integrated circuit (ASIC). In some other instances, the key generation module 118 may be implemented using a combination of hardware and software. Implementation in software may include rapid activation and deactivation of one or more transistors or transistor elements such as may be included in hardware of a computing system (e.g., the trusted entity server 108, the authentication server 140, and the user computing system 104). Additionally, software defined instructions may operate on information within transistor elements. Implementation of software instructions may at least temporarily reconfigure electronic pathways and transform computing hardware.

The database 116 may include any memory or data storage. The database 116 may include network communication capabilities such that the environment components may communicate with the database 116. In some embodiments, the database 116 may include computer-readable storage media for carrying or having computer-executable instructions or data structures stored thereon. The computer-readable storage media may include any available media that may be accessed by a general-purpose or special-purpose computer, such as a processor. For example, the database 116 may include computer-readable storage media that may be tangible or non-transitory computer-readable storage media including Random Access Memory (RAM), Read-Only Memory (ROM), Electrically Erasable Programmable Read-Only Memory (EEPROM), Compact Disc Read-Only Memory (CD-ROM) or other optical disk storage, magnetic disk storage or other magnetic storage devices, flash memory devices (e.g., solid state memory devices), or any other storage medium which may be used to carry or store desired program code in the form of computer-executable instructions or data structures and that may be accessed by a general-purpose or special-purpose computer. Combinations of the above may be included in the database 116.

In the depicted embodiment, the database 116 is separate from the authentication server 140. In some embodiments, the database 116 may be included in the authentication server 140 or the authentication server 140 may include a local database similar to the database 116 and may access the database 116 via the network 107.

The user 106 may include an individual or another entity such as a corporate entity, governmental entity, or the like. The user 106 may be associated with the user computing system 104 in some embodiments. For example, the user 106 may own or regularly operate the user computing system 104. In some embodiments, the user 106 may not be specifically associated with the user computing system 104. For example, the user computing system 104 may be publically accessible to multiple users including the user 106.

The user 106 may use the user computing system 104 to provide input to the encryption module 112. For example, the user 106 may operate a component of the user computing system 104 to provide input to the encryption module 112. The input may include one or more plaintext data sets that may be communicated or encrypted and communicated to the authentication module 114 of the authentication server 140.

The user computing system 104 may include a processor-based computing system. The user computing system 104 may include memory, a processor, and network communication capabilities. In the operating environment 100, the user computing system 104 may be capable of communicating data and information to the authentication server 140 via the network 107. Some examples of the user computing system 104 may include a mobile phone, a smartphone, a tablet computer, a laptop computer, a desktop computer, a set-top box, or a connected device (e.g., a smartwatch, smart glasses, a smart pedometer, or any other network-connectable device).

The user computing system 104 may include the encryption module 112. The encryption module 112 may be configured to encrypt plaintext data sets according to the relational encryption protocol using one or more keys. The encryption module 112 may be configured to receive the one or more keys from the trusted entity server 108, which one or more keys may be generated at the key generation module 118. Additionally, the encryption module 112 may be configured to access one or more keys that may be published.

The encryption module 112 may further receive input from the user 106 that includes plaintext data sets (e.g., a password). Using one or more of the keys, the encryption module 112 may encrypt the plaintext data sets and communicate the encrypted plaintext data sets to the authentication server 140. In some embodiments, the encrypted plaintext data sets may include a registration ciphertext and/or an encrypted response.

In addition, the encryption module 112 may be configured to receive a safeguard data set that may include a random challenge and/or an authentication signal communicated from the authentication server 140. The encryption module 112 may encrypt a plaintext data set using the safeguard data set and one or more of the keys. The authentication signal may reflect whether an authentication performed at the authentication server 140 is successful or unsuccessful. Based on the authentication signal, one or more processes or operations may be unlocked. For example, based on and/or in response to the authentication signal, the user 106 may access, modify, provide, etc. information and data to the user computing system 104 and/or another system (not shown).

The encryption module 112 may be implemented using hardware including a processor, a microprocessor (e.g., to perform or control performance of one or more operations), an FPGA, or an ASIC. In some other instances, the encryption module 112 may be implemented using a combination of hardware and software. Implementation in software may include rapid activation and deactivation of one or more transistors or transistor elements such as may be included in hardware of a computing system (e.g., the trusted entity server 108, the authentication server 140, and the user computing system 104). Additionally, software defined instructions may operate on information within transistor elements. Implementation of software instructions may at least temporarily reconfigure electronic pathways and transform computing hardware.

The authentication server 140 may include a processor-based computing device. For example, the authentication server 140 may include a hardware server or another processor-based computing device configured to function as a server. The authentication server 140 may include memory and network communication capabilities. In the operating environment 100, the authentication server 140 may be configured to communicate with the user computing system 104, the trusted entity server 108, and the database 116 via the network 107.

The authentication server 140 may include the authentication module 114. The authentication module 114 may be configured to verify equality using relational encryption. For example, the authentication module 114 may be configured to receive one or more keys from the trusted entity server 108. Additionally, the authentication module 114 may be configured to access one or more keys that may be published.

The authentication module 114 may receive a registration ciphertext from the user computing system 104 via the network 107. The authentication module 114 may store the registration ciphertext to the database 116. The authentication module 114 may not decrypt or have the capability to decrypt the registration ciphertext. In addition, the authentication module 114 may receive an authentication request, in response to which the authentication module 114 may communicate a safeguard data set that may include a random challenge. The authentication module 114 may then receive an encrypted response.

Using a relational encryption protocol, the authentication module 114 may verify equality between a first plaintext data set represented by the stored registration ciphertext and a second plaintext data set represented by the encrypted response. The equality between the first plaintext data set and the second plaintext data set may be based on a relationship between the stored registration ciphertext and the encrypted response. Based on whether equality exists between the first plaintext data set and the second plaintext data set, an authentication signal may be generated and communicated to the user computing system 104. For example, if equality exists, then the authentication signal may include an authentication message. If equality does not exist, then the authentication signal may include a fail message.

The authentication module 114 may be implemented using hardware including a processor, a microprocessor (e.g., to perform or control performance of one or more operations), an FPGA, or an ASIC. In some other instances, the authentication module 114 may be implemented using a combination of hardware and software. Implementation in software may include rapid activation and deactivation of one or more transistors or transistor elements such as may be included in hardware of a computing system (e.g., the trusted entity server 108, the authentication server 140, and the user computing system 104). Additionally, software defined instructions may operate on information within transistor elements. Implementation of software instructions may at least temporarily reconfigure electronic pathways and transform computing hardware.

Modifications, additions, or omissions may be made to the operating environment 100 without departing from the scope of the present disclosure. Specifically, the operating environment may include one or more users 106, one or more user computing systems 104, one or more authentication servers 140, one or more trusted entity servers 108, one or more databases 116, or any combination thereof. For example, the operating environment 100 may include another system with which the user computing system 104 interacts based on the authentication signal.

Moreover, the separation of various components in the embodiments described herein is not meant to indicate that the separation occurs in all embodiments. It may be understood with the benefit of this disclosure that the described environment components may be integrated together in a single component or separated into multiple components. For example, in some embodiments, the encryption module 112 and/or one or more functionalities attributed thereto may be performed by a module on the authentication server 140.

In the operating environment 100, memory in one or more of the environment components may be similar to memory 308 described with reference to FIG. 3, processors in one or more of the environment components may be similar to a processor 304 described with reference to FIG. 3, and network communication capabilities of one or more of the environment components may be provided by a communication unit such as a communication unit 302 described with reference to FIG. 3.

FIG. 2 illustrates an example equality verification process 200 that may be implemented in the operating environment 100 of FIG. 1. In the operating environment 100 of FIG. 2, the network 107 is omitted. Nevertheless, communication of information and data (e.g., 202, 204, 210, 216, 212, 214, and 218) between the trusted entity server 108, the user computer system 104, the authentication server 140, and the database 116 may be via the network 107. In general, the equality verification process 200 may include communication of one or more key sets (202 and 204) to the user computing system 104 and the authentication server 140. The user computing system 104 may receive a registration input 206, encrypt it as a registration ciphertext 210, and communicate the registration ciphertext 210 to the authentication server 140. The authentication server 140 may store the registration ciphertext 210 in the database 116. Subsequently, the user computing system 104 may communicate an authentication request 216 (in FIG. 2, “request 216”) to the authentication server 140. In response, the authentication server 140 may communicate a safeguard data set 212 that may include a random challenge to the user computing system 104. The user computing system 104 may receive the safeguard data set 212 and an authentication test input 208 (in FIG. 2, “test input 208”), encrypt the authentication test input 208 using the safeguard data set 212 as an encrypted response 214, and communicate the encrypted response 214 to the authentication server 140. The authentication server 140 may access the registration ciphertext 210. Using one or more of the keys, the authentication server 140 determines whether one or more relationships exist between the registration ciphertext 210 and the encrypted response 214. The relationships may indicate whether equality exists between the registration input 206 and the authentication test input 208 (e.g., if such relationships are present, then the equality exists) and thus may be used to verify equality between the registration input 206 and the authentication test input 208. The verification may be performed without decrypting the registration ciphertext 210 or the encrypted response 214. Based on whether equality exists between the registration input 206 and the authentication test input 208, the authentication server 140 may communicate an authentication signal 218 to the user computing system 104 or another system. Some additional details of the process 200 are provided in the following paragraphs.

The key generation module 118 of the trusted entity server 108 may be configured to generate keys used in a relational encryption protocol implemented in the equality verification process 200. In the depicted embodiment, the keys include a public key set 202 and a relational key 204. The relational key 204 may include a first relational key component and a secret hash key of a hash function. The public key set 202 may include a first public key, a second public key, a public hash key of the hash function, and the hash function.

To generate the keys, the key generation module 118 may generate bilinear groups. For example, given a particular security parameter, the bilinear groups may be generated of a prime order. The bilinear groups are exponential in the security parameter. The bilinear groups include a bilinear pairing operator. In some embodiments, the bilinear groups may be generated according to bilinear expression G1, G2, GT (x, q, e). In the bilinear expression, G1, G2, and GT represent bilinear groups. The parameter q represents the prime order. The parameter x represents the security parameter x. The parameter e represents the bilinear pairing operator. Some additional details of bilinear groups are discussed in U.S. patent application Ser. No. 14/287,051, which is incorporated herein by reference in its entirety.

The key generation module 118 may sample random sets from the bilinear groups. For example, one or more group elements may be randomly sampled from a first bilinear group of the bilinear groups and from a second bilinear group of the bilinear groups. In some embodiments, two group elements may be randomly sampled from the first bilinear group and three group elements may be randomly sampled from the second bilinear group. The two group elements of the first bilinear group may be related by a first group exponent. Additionally, two of the three group elements of the second bilinear group may be related by a second group exponent.

In some embodiments, the group elements may be randomly sampled according to bilinear group random sample expressions: (g1,g2)←G1; (h1,h2,A)←G2; g2=g1^(a); and h2=h1^(b). In the bilinear group random sample expressions, G1 and G2 represent a first bilinear group and a second bilinear group, respectively. The parameter g1 represents a first element of the first bilinear group. The parameter g2 represents a second element of the first bilinear group. The parameter h1 represents a first element of the second bilinear group. The parameter h2 represents a second element of the second bilinear group. The parameter A represents a third element of the second bilinear group. The parameter a represents a first group exponent. The parameter b represents a second group exponent.

The key generation module 118 may define one or more hash keys. The hash keys may include a secret hash key of a hash function and a public hash key of the hash function. The secret hash key may include a projected hash key including four secret hash key elements. The secret hash key elements may be randomly sampled from a set of integers of an order of the prime order of the bilinear groups. In some embodiments, the secret hash key may be defined according to secret hash key expressions: K=(k1,k2,k3,k4); and (k1,k2,k3,k4)∈Zq ⁴

In the secret hash key expressions, q is as described above. The parameter K represents the secret hash key. The parameters k1, k2, k3, and k4 represent secret hash key elements of the secret hash key. The operator ϵ is a membership operator. The symbol Zq⁴ represents a set of integers of an order of the prime order of the bilinear groups. The 4 in the symbol Zq⁴ indicates that four secret hash key elements k1, k2, k3, and k4 are sampled from the set of integers Zq⁴.

The public hash key may include a first public hash key element and a second public hash key element. The first public hash key element may include a product of a first group element of the second bilinear group raised to a power of a first secret hash key element and a third group element of the second bilinear group raised to a power of a second secret hash key element.

The second public hash key element may include a product of the first group element of the second bilinear group raised to a power of a third secret hash key element and the third group element of the second bilinear group raised to a power of a fourth secret hash key element.

In some embodiments, the public hash key may be defined according to a public hash key expression: S=(d,E)=(h1^(k1) A ^(k2) ,h1^(k3) A ^(k4)) In the public hash key expression, h1, k1, A, k2, k3, and k4 are as described above. The parameter S represents the public hash key. The parameter d represents the first public hash key element. The parameter E represents the second public hash key element.

The key generation module 118 may define the relational key 204. The relational key 204 may include a first relational key component and the secret hash key. The first relational key component may include a product of the second group exponent and an inverse of the first group exponent. For example, in some embodiments, the relational key 204 may be defined according to relational key expressions: pkR=ba ⁻¹; and rk:=(pkR,K).

In the relational key expressions, b, a, and K are as described above. The parameter pkR represents the first relational key component. The parameter rk represents the relational key.

The key generation module 118 may define the public key set 202. The public key set 202 may include a first public key, a second public key, the public hash key of the hash function, and the hash function. The first public key may include the first group element of the first bilinear group and the second group element of the first bilinear group. The second public key may include the first group element of the second bilinear group and the second group element of the second bilinear group. The hash function may include a collision resistant hash function.

In some embodiments, the public key set 202 may be defined according to public key set expressions: pk1=(g1,g2); pk2=(h1,h2); pk:=(pk1,pk2,S,HASH); and HASH:{0,1}*→Zq. In the public key set expressions, g1, g2, h1, h2, Zq, and S are as described above. The parameter pk1 represents the first public key. The parameter pk2 represents the second public key. The parameter pk represents the public key set. The operator := represents a definitional operator. The function HASH represents the hash function. Some examples of the hash function may include a SHA3, MD5, or another suitable collision resistant cryptographic hash function. The {0, 1}* represents an input to the hash function.

In the depicted embodiment, the public key set 202 may be communicated to the user computing system 104 and the relational key 204 may be communicated to the authentication server 140. In some embodiments, one or both of the public key set 202 and the relational key 204 may be published to a public site, which may be accessed by the user computing system 104 and/or the authentication server 140.

The registration input 206 may be generated and communicated to the user computing system 104. With combined reference to FIGS. 1 and 2, the user 106 may generate the registration input 206, which may be input to the user computing system 104 via a user input device, for instance.

The registration input 206 may include a first plaintext data set. The registration input 206 may be a member of the set of integers of an order of the prime order of the bilinear groups. Additionally, the registration input 206 may come from a low entropy distribution. Some examples of the registration input 206 may include a password, a private tag, a social security number, a birth date, or a credit card number. Details of the particular registration input 206 may depend on a type of the operating environment 100.

For example, the operating environment 100 may include a consumer application, an enterprise data repository, an application hosted on a cloud, another suitable password-protected application, or some combination thereof. In these types of operating environments, the registration input 206 may include a password. Additionally or alternatively, the operating environment 100 may include index database records. In these operating environments, the registration input 206 may include private tags used for index records in the index database records.

The encryption module 112 may be configured to encrypt the registration input 206. In some embodiments, a pseudo random generator (PRG) may be applied to the registration input 206. The PRG may be applied prior to encryption by the encryption module 112 of the registration input 206. The encrypted registration input 206 or the encrypted registration input 206 with the PRG applied may be the registration ciphertext 210.

Encryption by the encryption module 112 may include sampling a random registration sample. The random registration sample may be sampled from the set of integers of the order of the prime order of the bilinear groups.

The registration ciphertext 210 that results from encryption of the registration input 206 may include a first registration element and a second registration element. The first registration element may include the first group element of the first bilinear group raised to a power of the random registration sample. The second registration element may include a second group element of the first bilinear group raised to a product of the random registration sample and the registration input 206 (a first plaintext data set). For example, the registration ciphertext 210 may be defined according to registration ciphertext expressions: pwdϵZq; r←Zq; and (I1,I2):=(g1^(r) ,g2^(r·pwd)). In the registration ciphertext expressions, Zq, :=, ϵ, g1, and g2 are as discussed above. The parameter pwd represents the registration input 206, which may include a first plaintext data set. The parameter r represents the random registration sample. The parameter I1 represents the first registration element. The parameter I2 represents the second registration element.

The user computing system 104 may communicate the registration ciphertext 210 to the authentication server 140, which may store the registration ciphertext 210 in the database 116, a non-transitory computer-readable storage media, some combination thereof, or another suitable storage device.

The authentication server 140 may communicate the safeguard data set 212 to the user computing system 104. The safeguard data set 212 may include a random challenge that is a member of the set of integers of an order of the prime order of the bilinear groups.

The encryption module 112 may compute the encrypted response 214. The encrypted response 214 may include an encrypted representation of the authentication test input 208. For example, the encrypted response 214 may be based at least partially on the safeguard data set 212, the public hash keys, a random test sample data set, and the authentication test input 208, which may include a second plaintext data set.

In some circumstances the authentication test input 208 may be substantially similar or substantially equal to the registration input 206. For example, in some embodiments, the user computing system 104 may be configured to allow access to some information pending verification of a user password. The user password may be initially input as registration input 206. If the same user later wants access to the information, the user may later input the authentication test input 208. Assuming the user has not forgotten their password, the registration input 206 and the authentication test input 208 may be the same. If, however, an imposter attempts to gain access to the information, the imposter may input an authentication test input 208, which may not be the same as the registration input 206. In this example, a function of the user computing system 104, the authentication server 140, the trusted entity server 108, or some combination thereof may be to determine whether it is an imposter or the user attempting to gain access to the information.

The encrypted response 214 may include a first response element, a second response element, a third response element, and a hash proof. One or more of the first response element, the second response element, the third response element, and the hash proof may be computed based on a random test sample that is sampled from the set of integers of the order of the prime order of the bilinear groups.

The first response element may include the first group element of the second bilinear group raised to a power of the random test sample. The second response element may include the second group element of the second bilinear group raised to a power of a product of the random test sample and the second plaintext data set. The third response element may include the third group element of the second bilinear group raised to a power of the random test sample. The hash proof may include a product of the first public hash key element and the second public hash key element raised to a power of a response function. This product may then be raised to a power of the random test sample. The response function may include the hash function applied to the first response element, the second response element, the third response element, and the safeguard data set 212.

For example, in some embodiments, the encrypted response 214 may be computed according to encrypted response expressions: (H1,H2,B)=(h1^(s) ,h2^(s·pwd′) ,A ^(s)); f=HASH(H1,H2,B,t); HP=(de ^(f))^(s); pwd′ϵZq; s←Zq; and t←Zq In the encrypted response expressions, h1, h2, A, d, e, ϵ, and Zq are as described above. The parameter H1 represents the first response element. The parameter H2 represents the second response element. The parameter B represents the third response element. The value t represents the safeguard data set 212. The value s represents the random test sample. The parameter pwd′ represents the authentication test input 208. The parameter HP represents the hash proof. The function f represents the response function.

The user computing system 104 or a component thereof may communicate the encrypted response 214 to the authentication server 140. The encrypted response may be received at the authentication server 140. The authentication server 140 may compute a hash of the encrypted response 214.

The authentication server 140 may verify the hash proof. In some embodiments, verifying the hash proof includes determining whether the hash proof satisfies proof verification expression. In some embodiments, the hash proof verification expression may include: HP

H1^(k1+fk2) A ^(k2+fk4) In the hash proof verification expression, HP, H1, k1, f, k2, A, and k4 are as described above.

The authentication module 114 may verify whether equality exists between the registration input 206 and the authentication test input 208. Equality may exist between the registration input 206 and the authentication test input 208 when the registration input 206 and the authentication test input 208 are the same. The registration input 206 and the authentication test input 208 may be the same in circumstances in which a user (e.g., the user 106 of FIG. 1) generates the registration input 206 and the authentication test input 208. Equality may not exist between the registration input 206 and the authentication test input 208 when the registration input 206 and the authentication test input 208 are different. The difference between the registration input 206 and the authentication test input 208 may be indicative of an imposter inputting the authentication test input 208, for example.

Verification of the equality may be performed without decrypting the encrypted response 214 or the registration ciphertext 210. The verification may be based on a relationship between the encrypted response 214 and the registration ciphertext 210 as ascertained using the relational key 204. In some embodiments, verifying whether the equality exists includes determining whether an equality expression is satisfied. In some embodiments, the equality expression includes: e(I1,H2)

e(I2,H1)^(pkR) In the equality expression e, I1, H2, 12, H1 and pkR are as described above.

The authentication signal 218 may be indicative of whether there is equality between the registration input 206 and the authentication test input 208 and whether the hash proof is verified. For example, if there is equality between the registration input 206 and the authentication test input 208 and the hash proof is verified, then the authentication signal 218 may include an authentication message. If there is not equality between the registration input 206 and the authentication test input 208 or the hash proof is not verified, the authentication signal 218 may include a fail message.

FIG. 3 illustrates an example computing system 300 configured for equality verification using relational encryption. The computing system 300 may be implemented in the operating environment 100. Examples of the computing system 300 may include one or more of the user computing system 104, the authentication server 140, and the trusted entity server 108.

The computing system 300 may include one or more processors 304, a memory 308, a communication unit 302, a user input device 314, and a data storage 301 that further includes one or more of the authentication module 114, the key generation module 118, and the encryption module 112 (collectively, modules 112/114/118).

The processor 304 may include any suitable special-purpose or general-purpose computer, computing entity, or processing device including various computer hardware or software modules and may be configured to execute instructions stored on any applicable computer-readable storage media. For example, the processor 304 may include a microprocessor, a microcontroller, a digital signal processor (DSP), an ASIC, an FPGA, or any other digital or analog circuitry configured to interpret and/or to execute program instructions and/or to process data.

Although illustrated as a single processor in FIG. 3, the processor 304 may more generally include any number of processors configured to perform individually or collectively any number of operations described in the present disclosure. Additionally, one or more of the processors 304 may be present on one or more different electronic devices or computing systems. In some embodiments, the processor 304 may interpret and/or execute program instructions and/or process data stored in the memory 308, the data storage 301, or the memory 308 and the data storage 301. In some embodiments, the processor 304 may fetch program instructions from the data storage 301 and load the program instructions in the memory 308. After the program instructions are loaded into the memory 308, the processor 304 may execute the program instructions.

The memory 308 and the data storage 301 may include computer-readable storage media for carrying or having computer-executable instructions or data structures stored thereon. Such computer-readable storage media may include any available media that may be accessed by a general-purpose or special-purpose computer, such as the processor 304. By way of example, and not limitation, such computer-readable storage media may include tangible or non-transitory computer-readable storage media including RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, flash memory devices (e.g., solid state memory devices), or any other storage medium which may be used to carry or store desired program code in the form of computer-executable instructions or data structures and that may be accessed by a general-purpose or special-purpose computer. Combinations of the above may also be included within the scope of computer-readable storage media. Computer-executable instructions may include, for example, instructions and data configured to cause the processor 304 to perform a certain operation or group of operations.

The communication unit 302 may include one or more pieces of hardware configured to receive and send communications. In some embodiments, the communication unit 302 may include one or more of an antenna, a wired port, and modulation/demodulation hardware, among other communication hardware devices. In particular, the communication unit 302 may be configured to receive a communication from outside the computing system 300 and to present the communication to the processor 304 or to send a communication from the processor 304 to another device or network.

The user input device 314 may include one or more pieces of hardware configured to receive input from and/or provide output to a user. In some embodiments, the user input device 314 may include one or more of a speaker, a microphone, a display, a keyboard, and a touch screen, a holographic projection, among other hardware devices. In these and other embodiments, the user input device 314 may be configured to receive input from a user (e.g., the user 106 of FIG. 1) of the computing system 300.

The modules 112/114/118 may include program instructions stored in the data storage 301. The processor 304 may be configured to load the modules 112/114/118 into the memory 308 and execute the modules 112/114/118. Alternatively, the processor 304 may execute the modules 112/114/118 line-by-line from the data storage 301 without loading them into the memory 308. When executing the modules 112/114/118, the processor 304 may be configured for equality verification using relational encryption as described elsewhere herein.

Modifications, additions, or omissions may be made to the computing system 300 without departing from the scope of the present disclosure. For example, in some embodiments, the computing system 300 may not include the user input device 314. In some embodiments, the different components of the computing system 300 may be physically separate and may be communicatively coupled via any suitable mechanism. For example, the data storage 301 may be part of a storage device that is separate from a server, which includes the processor 304, the memory 308, and the communication unit 302, that is communicatively coupled to the storage device.

FIG. 4 is a flow diagram of an example method 400 of equality verification using relational encryption, arranged in accordance with at least one embodiment described herein. The method 400 may be performed in an operating environment such as the operating environment 100 of FIGS. 1 and 2. The method 400 may be programmably performed in some embodiments by some combination of the trusted entity server 108, the authentication server 140, and the user computing system 104 described with reference to FIGS. 1 and 2. The trusted entity server 108, the authentication server 140, and the user computing system 104 may include or may be communicatively coupled to a non-transitory computer-readable medium (e.g., the memory 308 or data storage 301 of FIG. 3) having stored thereon or encoded therein programming code or instructions that are executable by a processor to perform or cause performance of the method 400. Additionally or alternatively, the trusted entity server 108, the authentication server 140, and the user computing system 104 may include a processor (e.g., the processor 304 of FIG. 3) that is configured to execute computer instructions to perform or control performance of the method 400. Although illustrated as discrete blocks, various blocks may be divided into additional blocks, combined into fewer blocks, or eliminated, depending on the desired implementation.

Portions of the method 400 may be performed by one or more systems. For example, in the depicted flow chart, blocks 402, 404, 406, 408, 410, and 412 may be performed by a trusted entity server such as the trusted entity server 108 or a component thereof. Additionally, blocks 414, 416, 418, 420, 426, and 428 may be performed by a user computing systems such as the user computing system 104 or a component thereof and blocks 422, 423, 424, 429, 430, 432, 434, and 436 may be performed by an authentication server such as the authentication server 140 or a component thereof. In some embodiments one or more portions of the method 400 may be performed by another of the systems 108, 104, 140 or by one or more other systems.

The method 400 may begin at block 402 in which bilinear groups may be generated. In the depicted flow chart, the trusted entity server 108 may generate the bilinear groups. For example, given a particular security parameter, bilinear groups may be generated of a prime order. The bilinear groups may be exponential in the security parameter. The bilinear groups may include a bilinear pairing operator.

At block 404, one or more group elements may be sampled from one or more of the bilinear groups. For example, two group elements may be sampled from a first bilinear group of the bilinear groups and three group elements may be sampled from a second bilinear group of the bilinear groups. The two group elements sampled from the first bilinear group may be related by a first group exponent. Additionally, two of the three group elements sampled from the second bilinear group may be related by a second group exponent.

At block 406, one or more hash keys may be defined. The hash keys may include a secret hash key of a hash function and a public hash key of the hash function. The secret hash key may include a projected hash key including four secret hash key elements that are random samples of a set of integers of an order of the prime order of the bilinear groups.

The public hash key may include a first public hash key element and a second public hash key element. For example, the public hash key may include the first public hash key element and the second hash key element discussed with reference to FIG. 2.

At block 408, a relational key may be defined. The relational key may include a first relational key component and the secret hash key. The first relational key component includes a product of the second group exponent and an inverse of the first group exponent.

At block 410, a public key set may be defined. The public key set may include a first public key, a second public key, the public hash key of the hash function, and the hash function. The first public key may include the group elements sampled from the first bilinear group. The second public key may include two of the three group elements of the second bilinear group. The hash function may include a collision resistant hash function.

At block 412, the public key set and the relational key may be communicated to the user computing system 104 and/or the authentication server 140. For example, the authentication server 140 may be given access to the relational key and the public key set. The public key set may be communicated to the user computing system 104.

At block 414, a first plaintext data set may be received. The first plaintext data set may be a member of the set of integers of an order of the prime order of the bilinear groups and/or may come from a low entropy distribution. The first plaintext data set may include a password such as a password implemented in a consumer application, enterprise data repository, an application hosted on a cloud, a combination thereof or another suitable password-protected application. The first plaintext data set may also include a private tag, which may be used to index database records or a social security number, a birthdate, a credit card number, or the like. In some embodiments, a pseudo random generator (PRG) may be applied to the first plaintext data set. The PRG may be applied prior to encryption of the first plaintext data set.

At block 416, a random registration sample may be sampled. The random registration sample may be sampled from the set of integers of the order of the prime order of the bilinear groups. At block 418, the first plaintext data set may be encrypted. The encryption of the first plaintext data set may be a registration ciphertext. The registration ciphertext includes a first registration element and a second registration element. The first registration element may be similar to the first registration element and the second registration element discussed with reference to FIG. 2.

At block 420, the registration ciphertext may be communicated. For example, the registration ciphertext may be communicated to the authentication server 140. At block 422, the registration ciphertext may be stored. For example, the authentication server 140 may store the registration ciphertext on a database such as the database 116 of FIGS. 1 and 2. The registration ciphertext may include the encryption of the first plaintext data set.

At block 423, an authentication request may be received. The authentication request may be received from the user computing system 104. At block 424, a safeguard data set that may include a random challenge may be communicated. For example, the safeguard data set may be communicated to the user computing system 104 in response to the authentication request being received. The random challenge may be a member of the set of integers of an order of the prime order of the bilinear groups.

At block 426, an encrypted response may be computed. The encrypted response may be computed that includes an encrypted representation of a second plaintext data set. For example, the encrypted response may include a first response element, a second response element, a third response element, and a hash proof. One or more of the first response element, the second response element, the third response element, and the hash proof may be computed as described with reference to FIG. 2. At block 428, the encrypted response may be communicated to the authentication server 140.

At block 429, the encrypted response may be received. Generally, the encrypted response may be received after the registration ciphertext is stored. At block 430, a hash of the received encrypted response may be computed. At block 432, the hash proof may be verified. In some embodiments, verifying the hash proof includes determining whether the hash proof satisfies a hash proof verification expression such as the hash proof verification expression discussed with reference to FIG. 2.

At block 434, equality may be verified. For example, it may be verified whether equality exists between the first plaintext data set and the second plaintext data set. Verification of the equality may be performed without decrypting the encrypted response or the registration ciphertext. The verification may be based on a relationship between the encrypted response and the registration ciphertext as ascertained using the relational key. In some embodiments, verifying whether the equality exists includes determining whether an equality expression is satisfied. For example, the equality expression may verify that the bilinear pairing operator applied to the first registration element and to the second response element is equal to the bilinear pairing operator applied to the second registration element and the first response element all taken to a power of the first relational key component.

At block 436, an authentication signal may be communicated to the user computing system 104. The authentication signal may be indicative of whether there is equality between the first and second plaintext data sets and/or whether the hash proof is verified. For example, if there is equality between the first and the second plaintext data sets and the hash proof is verified, then the authentication signal may include an authentication message. If there is not equality between the first and the second plaintext data sets or the hash proof is not verified, the authentication signal may include a fail message.

One skilled in the art will appreciate that, for this and other procedures and methods disclosed herein, the functions performed in the processes and methods may be implemented in differing order. Furthermore, the outlined steps and operations are only provided as examples, and some of the steps and operations may be optional, combined into fewer steps and operations, or expanded into additional steps and operations without detracting from the disclosed embodiments.

FIG. 5 is a flow diagram of an example method 500 of equality verification using relational encryption, arranged in accordance with at least one embodiment described herein. The method 500 may be performed in an operating environment such as the operating environment 100 of FIGS. 1 and 2. The method 500 may be programmably performed in some embodiments by the authentication server 140 described with reference to FIGS. 1 and 2. The authentication server 140 may include or may be communicatively coupled to a non-transitory computer-readable medium (e.g., the memory 308 of FIG. 3) having stored thereon or encoded therein programming code or instructions that are executable by a processor to perform or cause performance of the method 500. Additionally or alternatively, the authentication server 140 may include a processor (e.g., the processor 304 of FIG. 3) that is configured to execute computer instructions to perform or control performance of the method 500. Although illustrated as discrete blocks, various blocks may be divided into additional blocks, combined into fewer blocks, or eliminated, depending on the desired implementation.

The method 500 may begin at block 502 in which a relational key is received. The relational key may include a first relational key component and a secret hash key of a hash function. In some embodiments, the secret hash key includes a projected hash key including four secret hash key elements that may include random samples of a set of integers of an order of a prime order of bilinear groups. The bilinear groups may be generated based on a particular security parameter. The bilinear groups may be of a prime order. The bilinear groups may be exponential in the particular security parameter. The bilinear groups may include a bilinear pairing operator.

At block 504, a public key set may be received. The public key set may include a first public key, a second public key, a public hash key of the hash function, and the hash function.

The first public key includes two group elements randomly selected from a first bilinear group of the bilinear groups. The second public key includes two group elements of a second bilinear group of the bilinear groups. The two group elements of the first bilinear group are randomly sampled from the first bilinear group. The two group elements of the second bilinear group are randomly sampled from the second bilinear group.

In some embodiments, the two group elements of the first public key may be related by a first group exponent. The two group elements of the second public key may be related by a second group exponent. In these and other embodiments, the first relational key component includes a product of the second group exponent and an inverse of the first group exponent.

In some embodiments, the public hash key includes a first public hash key group element and a second public hash key group element. The first public hash key group element includes a product of a first group element of the two group elements of the second public key raised to a power of a first secret hash key element of the four secret hash key elements and a third group element of the second bilinear group raised to a power of a second secret hash key element of the four secret hash key elements.

The second public hash key group element includes a product of the first group element of the second bilinear group raised to a power of a third secret hash key element and the third group element of the second bilinear group raised to a power of a fourth secret hash key element.

At block 506, a registration ciphertext may be received, e.g., from a user computing system. The registration ciphertext may include an encryption of a first plaintext data set. The first plaintext data set may be representative of a password of a particular user, for instance. The registration ciphertext may include a first registration element and a second registration element. The first registration element may include the first group element of the first bilinear group raised to a power of a random registration sample. The second registration element may include a second group element of the first bilinear group raised to a product of the random registration sample and the first plaintext data set.

At block 508, the registration ciphertext may be stored. The registration ciphertext may be stored without decrypting the registration ciphertext. At block 510, an authentication request may be received, e.g., from a user computing system. In some embodiments, after the storing of the registration ciphertext, the authentication request may be received. At block 512, a safeguard data set that may include a random challenge may be communicated. For example, the safeguard data set may be communicated in response to the authentication request. The random challenge may be a member of a set of integers of an order of the prime order of the bilinear groups.

At block 514, an encrypted response may be received, e.g., from a user computing system. The encrypted response may be generated based at least partially on the safeguard data set and a second plaintext data set. The encrypted response may be further based on one or more public hash key elements of the public hash key and a random test sample data set. For example, the encrypted response may include a first response element, a second response element, a third response element, and a hash proof. The first response element may include the first group element of the second bilinear group raised to a power of a random encrypted response sample. The second response element may include the second group element of the second bilinear group raised to a power of a product of the encrypted response random sample and the second plaintext data set. The third response element may include the third group element of the second bilinear group raised to a power of the encrypted response random sample. The hash proof includes a value raised to a power of the encrypted response random sample. The value is a product of the first public hash key element and the second public hash key element raised to a power of a response function. The response function may include the hash function applied to the first response element, the second response element, the third response element, and the safeguard data set.

At block 516, the hash proof may be verified. In some embodiments, verifying the hash proof may include determining whether the hash proof satisfies a hash proof verification expression. An example of the hash proof verification expression is provided elsewhere in this disclosure.

At block 518, a relationship between the encrypted response and the registration ciphertext may be verified. The relationship may be verified using the relational key without decrypting the encrypted response and without decrypting the registration ciphertext. The relationship may indicate that equality exists between the first plaintext data set and the second plaintext data set. In some embodiments, the verifying the relationship includes determining whether an equality expression is satisfied. An example of the equality expression is provided elsewhere in this disclosure. At block 520, an authentication signal may be communicated. In some embodiments, the authentication signal is representative of whether equality exists between the first plaintext data set and the second plaintext data set.

The embodiments described herein may include the use of a special-purpose or general-purpose computer including various computer hardware or software modules, as discussed in greater detail below.

Embodiments described herein may be implemented using computer-readable media for carrying or having computer-executable instructions or data structures stored thereon. Such computer-readable media may be any available media that may be accessed by a general-purpose or special purpose computer. By way of example, and not limitation, such computer-readable media may include non-transitory computer-readable storage media including RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, flash memory devices (e.g., solid state memory devices), or any other storage medium which may be used to carry or store desired program code in the form of computer-executable instructions or data structures and which may be accessed by a general-purpose or special-purpose computer. Combinations of the above may also be included within the scope of computer-readable media.

Computer-executable instructions comprise, for example, instructions and data which cause a general-purpose computer, special-purpose computer, or special-purpose processing device (e.g., one or more processors) to perform a certain function or group of functions. Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.

As used herein, the terms “module” or “component” may refer to specific hardware implementations configured to perform the operations of the module or component and/or software objects or software routines that may be stored on and/or executed by general-purpose hardware (e.g., computer-readable media, processing devices, etc.) of the computing system. In some embodiments, the different components, modules, engines, and services described herein may be implemented as objects or processes that execute on the computing system (e.g., as separate threads). While some of the system and methods described herein are generally described as being implemented in software (stored on and/or executed by general purpose hardware), specific hardware implementations or a combination of software and specific hardware implementations are also possible and contemplated. In this description, a “computing entity” may be any computing system as previously defined herein, or any module or combination of modulates running on a computing system.

All examples and conditional language recited herein are intended for pedagogical objects to aid the reader in understanding the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Although embodiments of the present inventions have been described in detail, it should be understood that the various changes, substitutions, and alterations could be made hereto without departing from the spirit and scope of the invention. 

What is claimed is:
 1. A method of equality verification using relational encryption, the method comprising: receiving, from a trusted entity server, a relational key that includes a first relational key component; receiving, from a user computing system, a registration ciphertext that includes an encryption of a first plaintext data set; storing, in a non-transitory computer-readable storage medium, the registration ciphertext without decrypting the registration ciphertext; after the storing of the registration ciphertext, receiving, from the user computing system, an authentication request; in response to the authentication request, communicating a safeguard data set that includes a random challenge to the user computing system; receiving, from the user computing system, an encrypted response that is generated based at least partially on the safeguard data set and a second plaintext data set; and verifying, by one or more processors, a relationship between the encrypted response and the registration ciphertext using the relational key without decrypting the encrypted response and without decrypting the registration ciphertext, the relationship indicating that equality exists between the first plaintext data set and the second plaintext data set.
 2. The method of claim 1, further comprising receiving, from the trusted entity server, a public key set that includes a first public key, a second public key, a public hash key of a hash function, and the hash function, wherein: the relational key further includes a secret hash key of the hash function, and the encrypted response is further based on one or more public hash key elements of the public hash key and a random test sample data set.
 3. The method of claim 2, wherein: the first public key includes two group elements of a first bilinear group of multiple bilinear groups; the second public key includes two group elements of a second bilinear group of the bilinear groups; the two group elements of the first bilinear group are randomly sampled from the first bilinear group; the two group elements of the second bilinear group are randomly sampled from the second bilinear group; the bilinear groups are generated based on a particular security parameter; the bilinear groups are of a prime order; the bilinear groups are exponential in the particular security parameter; and the bilinear groups include a bilinear pairing operator.
 4. The method of claim 3, wherein: the two group elements of the first public key are related by a first group exponent; the two group elements of the second public key are related by a second group exponent; and the first relational key component includes a product of the second group exponent and an inverse of the first group exponent.
 5. The method of claim 4, wherein: the secret hash key includes a projected hash key including four secret hash key elements that are random samples of a set of integers of an order of the prime order of the bilinear groups; the public hash key includes a first public hash key group element and a second public hash key group element; the first public hash key group element includes a product of a first group element of the two group elements of the second public key raised to a power of a first secret hash key element of the four secret hash key elements and a third group element of the second bilinear group raised to a power of a second secret hash key element of the four secret hash key elements; and the second public hash key group element includes a product of the first group element of the second bilinear group raised to a power of a third secret hash key element and the third group element of the second bilinear group raised to a power of a fourth secret hash key element.
 6. The method of claim 5, wherein: the encrypted response includes a first response element, a second response element, a third response element, and a hash proof; the first response element includes a first group element of a second bilinear group of bilinear groups raised to a power of a random encrypted response sample; the second response element includes a second group element of the second bilinear group raised to a power of a product of the random encrypted response sample and the second plaintext data set; the third response element includes the third group element of the second bilinear group raised to a power of the random encrypted response sample; the hash proof includes a value raised to a power of the encrypted response random sample, wherein the value is a product of the first public hash key element and the second public hash key element raised to a power of a response function; and the response function includes the hash function applied to the first response element, the second response element, the third response element, and the safeguard data set.
 7. The method of claim 6, further comprising verifying the hash proof, wherein the verifying the hash proof includes determining whether the hash proof satisfies a hash proof verification expression: HP

H1^(k1+fk2) A ^(k2+fk4), in which: HP represents the hash proof; H1 represents the first response element; k1 represents the first secret hash key element; k2 represents the second secret hash key element; k3 represents the third secret hash key element; k4 represents the fourth secret hash key element; f represents the response function; and A represents a third group element of the second bilinear group.
 8. The method of claim 7, wherein: the verifying the relationship includes determining whether an equality expression is satisfied; and the equality expression includes: e(I1,H2)

e(I2,H1)^(pkR), in which: e represents a bilinear pairing operator; H2 represents the second response element; I1 represents a first registration element of the registration ciphertext; I2 represents a second registration element of the registration ciphertext; and pkR represents the first relational key component.
 9. The method of claim 1, wherein: the registration ciphertext includes a first registration element and a second registration element; the first registration element includes a first group element of a first bilinear group of bilinear groups raised to a power of a random registration sample; the second registration element includes a second group element of the first bilinear group raised to a product of the random registration sample and the first plaintext data set; the first group element and the second group element are random samples of the first bilinear group; the bilinear groups are generated based on a particular security parameter; the bilinear groups are of a prime order; the bilinear groups are exponential in the particular security parameter; and the bilinear groups include a bilinear pairing operator.
 10. The method of claim 9, wherein: the random challenge is a member of a set of integers of an order of the prime order of the bilinear groups; the first plaintext data set is representative of a password of a particular user; and the method further comprises communicating an authentication signal, wherein the authentication signal is representative of whether equality exists between the first plaintext data set and the second plaintext data set.
 11. A non-transitory computer-readable medium having encoded therein programming code executable by a processor to perform or control performance of operations comprising: receiving, from a trusted entity server, a relational key that includes a first relational key component; receiving, from a user computing system, a registration ciphertext that includes an encryption of a first plaintext data set; storing, in a non-transitory computer-readable storage media, the registration ciphertext without decrypting the registration ciphertext; after the storing of the registration ciphertext, receiving, from the user computing system, an authentication request; in response to the authentication request, communicating a safeguard data set that includes a random challenge to the user computing system; receiving, from the user computing system, an encrypted response that is generated based at least partially on the safeguard data set and a second plaintext data set; and verifying a relationship between the encrypted response and the registration ciphertext using the relational key without decrypting the encrypted response and without decrypting the registration ciphertext, the relationship indicating that equality exists between the first plaintext data set and the second plaintext data set.
 12. The non-transitory computer-readable medium of claim 11, wherein: the operations further comprise receiving a public key set that includes a first public key, a second public key, a public hash key of a hash function, and the hash function; the relational key further includes a secret hash key of the hash function; and the encrypted response is further based on one or more public hash key elements of the public hash key and a random test sample data set.
 13. The non-transitory computer-readable medium of claim 12, wherein: the first public key includes two group elements of a first bilinear group of multiple bilinear groups; the second public key includes two group elements of a second bilinear group of the bilinear groups; the two group elements of the first bilinear group are randomly sampled from the first bilinear group; the two group elements of the second bilinear group are randomly sampled from the second bilinear group; the bilinear groups are generated based on a particular security parameter; the bilinear groups are of a prime order; the bilinear groups are exponential in the particular security parameter; and the bilinear groups include a bilinear pairing operator.
 14. The non-transitory computer-readable medium of claim 13, wherein: the two group elements of the first public key are related by a first group exponent; the two group elements of the second public key are related by a second group exponent; and the first relational key component includes a product of the second group exponent and an inverse of the first group exponent.
 15. The non-transitory computer-readable medium of claim 14, wherein: the secret hash key includes a projected hash key including four secret hash key elements that are random samples of a set of integers of an order of the prime order of the bilinear groups; the public hash key includes a first public hash key group element and a second public hash key group element; the first public hash key group element includes a product of a first group element of the two group elements of the second public key raised to a power of a first secret hash key element of the four secret hash key elements and a third group element of the second bilinear group raised to a power of a second secret hash key element of the four secret hash key elements; and the second public hash key group element includes a product of the first group element of the second bilinear group raised to a power of a third secret hash key element and the third group element of the second bilinear group raised to a power of a fourth secret hash key element.
 16. The non-transitory computer-readable medium of claim 15, wherein: the encrypted response includes a first response element, a second response element, a third response element, and hash proof; the first response element includes a first group element of a second bilinear group of bilinear groups raised to a power of a random encrypted response sample; the second response element includes a second group element of the second bilinear group raised to a power of a product of the random encrypted response sample and the second plaintext data set; the third response element includes the third group element of the second bilinear group raised to a power of the random encrypted response sample; the hash proof includes a value raised to a power of the random encrypted response sample, wherein the value is a product is of the first public hash key element and the second public hash key element raised to a power of a response function; and the response function includes the hash function applied to the first response element, the second response element, the third response element, and the safeguard data set.
 17. The non-transitory computer-readable medium of claim 16, wherein: the operations further comprise verifying the hash proof; and the verifying the hash proof includes determining whether the hash proof satisfies a hash proof verification expression: HP

H1^(k1+fk2) A ^(k2+fk4), in which: HP represents the hash proof; H1 represents the first response element; k1 represents the first secret hash key element; k2 represents the second secret hash key element; k3 represents the third secret hash key element; k4 represents the fourth secret hash key element; f represents the response function; and A represents a third group element of the second bilinear group.
 18. The non-transitory computer-readable medium of claim 17, wherein: the verifying the relationship includes determining whether an equality expression is satisfied; and the equality expression includes: e(I1,H2)

e(I2,H1)^(pkR), in which: e represents a bilinear pairing operator; H2 represents the second response element; I1 represents a first registration element of the registration ciphertext; 12 represents a second registration element of the registration ciphertext; and pkR represents the first relational key component.
 19. The non-transitory computer-readable medium of claim 11, wherein: the registration ciphertext includes a first registration element and a second registration element; the first registration element includes a first group element of a first bilinear group of bilinear groups raised to a power of a random registration sample; the second registration element includes a second group element of the first bilinear group raised to a product of the random registration sample and the first plaintext data set; the first group element and the second group element are random samples of the first bilinear group; the bilinear groups are generated based on a particular security parameter; the bilinear groups are of a prime order; the bilinear groups are exponential in the particular security parameter; and the bilinear groups include a bilinear pairing operator.
 20. The non-transitory computer-readable medium of claim 19, wherein: the random quantity that is a member of a set of integers of an order of the prime order of the bilinear groups; the first plaintext data set is representative of a password of a particular user; the operations further comprise communicating an authentication signal; and the authentication signal is representative of whether equality exists between the first plaintext data set and the second plaintext data set. 